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| + | [[Category: ECE]] | ||
| + | [[Category: ECE 301]] | ||
| + | [[Category: Fall]] | ||
| + | [[Category: 2007]] | ||
| + | [[Category: mboutin]] | ||
| + | [[Category: Defintions]] | ||
| + | [[Category: Fourier]] | ||
| + | [[Category: Fourier Transform]] | ||
| + | [[Category: Tables]] | ||
| − | {| | + | {| class="wikitable" style="margin: 1em auto 1em auto" border="1" cellpadding="5" cellspacing="0" |
! Time Domain !! Fourier Domain | ! Time Domain !! Fourier Domain | ||
|- | |- | ||
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| <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t </math> | | <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t </math> | ||
|- | |- | ||
| − | | <math> 1\ </math> || <math> 2 \pi \delta (\omega) </math> | + | | <math> 1\ </math> || <math> 2 \pi \delta (\omega)\ </math> |
|- | |- | ||
| − | | <math> | + | | <math> -0.5+u(t)\ </math> || <math> \frac{1}{j \omega}\ </math> |
|- | |- | ||
| − | | <math> \delta (t) \ </math> || <math> 1\ </math> | + | | <math> \delta (t)\ </math> || <math> 1\ </math> |
|- | |- | ||
| − | | <math> \delta (t-c)\ </math> || <math> e | + | | <math> \delta (t-c)\ </math> || <math> e^{-j \omega c}\ </math> |
|- | |- | ||
| − | | <math> u(t) </math> || <math> \pi \delta(\omega)+\frac{1}{j \omega} </math> | + | | <math> u(t)\ </math> || <math> \pi \delta(\omega)+\frac{1}{j \omega} </math> |
|- | |- | ||
| − | | <math> e ^{ | + | | <math> e^{-bt}u(t)\ </math> || <math> \frac{1}{j \omega + b} </math> |
|- | |- | ||
| − | | <math> cos \omega_0 t </math> || <math> \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 )] </math> | + | | <math> cos \omega_0 t\ </math> || <math> \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 ) ]\ </math> |
|- | |- | ||
| − | |cos( | + | | <math> cos ( \omega_0 t + \theta )\ </math> |
| + | | <math> \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) + e^{j \theta} \delta ( \omega - \omega_0 )]\ </math> | ||
|- | |- | ||
| − | | | + | | <math> sin \omega_0 t\ </math> || <math> j \pi [ \delta ( \omega + \omega_0 ) - \delta ( \omega - \omega_0 ) ]\ </math> |
|- | |- | ||
| − | |sin( | + | | <math> sin ( \omega_0 t + \theta )\ </math> |
| + | | <math> j \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) - e^{j \theta} \delta ( \omega - \omega_0 ) ]\ </math> | ||
|- | |- | ||
| − | |rect(\frac{t}{\tau}) | + | | <math> rect \left ( \frac{t}{\tau} \right ) </math> || <math> \tau sinc \frac{\tau \omega}{2 \pi} </math> |
|- | |- | ||
| − | |\tau sinc \frac{\tau t}{2 \pi} | + | | <math> \tau sinc \frac{\tau t}{2 \pi} </math> || <math> 2 \pi p_\tau\ ( \omega ) </math> |
|- | |- | ||
| − | (1-\frac{2 |t|}{\tau})p_\tau (t) | + | | <math> \left ( 1-\frac{2 |t|}{\tau} \right ) p_\tau (t) </math> || <math> \frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi} </math> |
|- | |- | ||
| − | \frac{\tau}{2} sinc^2 ( \frac{\tau t}{4 \pi} ) | + | | <math> \frac{\tau}{2} sinc^2 \left ( \frac{\tau t}{4 \pi} \right ) </math> |
| + | | <math> 2 \pi \left ( 1-\frac{2|\omega|}{\tau} \right ) p_\tau (\omega) </math> | ||
|} | |} | ||
| − | |||
| − | + | Notes: | |
| + | : <math> sinc(x) = \frac {sin(x)}{x} </math> <br> | ||
| + | : <math> p_\tau (t)\ </math> is the rectangular pulse function of width <math> \tau\ </math> | ||
| + | |||
| + | |||
| + | Source courtesy Wikibooks.org | ||
Latest revision as of 11:28, 10 December 2008
| Time Domain | Fourier Domain |
|---|---|
| $ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega $ | $ X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t $ |
| $ 1\ $ | $ 2 \pi \delta (\omega)\ $ |
| $ -0.5+u(t)\ $ | $ \frac{1}{j \omega}\ $ |
| $ \delta (t)\ $ | $ 1\ $ |
| $ \delta (t-c)\ $ | $ e^{-j \omega c}\ $ |
| $ u(t)\ $ | $ \pi \delta(\omega)+\frac{1}{j \omega} $ |
| $ e^{-bt}u(t)\ $ | $ \frac{1}{j \omega + b} $ |
| $ cos \omega_0 t\ $ | $ \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 ) ]\ $ |
| $ cos ( \omega_0 t + \theta )\ $ | $ \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) + e^{j \theta} \delta ( \omega - \omega_0 )]\ $ |
| $ sin \omega_0 t\ $ | $ j \pi [ \delta ( \omega + \omega_0 ) - \delta ( \omega - \omega_0 ) ]\ $ |
| $ sin ( \omega_0 t + \theta )\ $ | $ j \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) - e^{j \theta} \delta ( \omega - \omega_0 ) ]\ $ |
| $ rect \left ( \frac{t}{\tau} \right ) $ | $ \tau sinc \frac{\tau \omega}{2 \pi} $ |
| $ \tau sinc \frac{\tau t}{2 \pi} $ | $ 2 \pi p_\tau\ ( \omega ) $ |
| $ \left ( 1-\frac{2 |t|}{\tau} \right ) p_\tau (t) $ | $ \frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi} $ |
| $ \frac{\tau}{2} sinc^2 \left ( \frac{\tau t}{4 \pi} \right ) $ | $ 2 \pi \left ( 1-\frac{2|\omega|}{\tau} \right ) p_\tau (\omega) $ |
Notes:
- $ sinc(x) = \frac {sin(x)}{x} $
- $ p_\tau (t)\ $ is the rectangular pulse function of width $ \tau\ $
Source courtesy Wikibooks.org
